AISM 53, 599-619

## Distributions of numbers of success runs of fixed length in Markov dependent trials

### Demetrios L. Antzoulakos and Stathis Chadjiconstantinidis

Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou Str., 18534 Piraeus, Greece, e-mail:dantz@unipi.gr

(Received January 11, 1999; revised August 9, 1999)

Abstract.    Let \$\{ Z_n, n\geq 1\}\$ be a time-homogeneous \$\{0,1\}\$-valued Markov chain, and let \$N_n\$ be a random variable denoting the number of runs of "\$1\$" of length \$k\$ in the first \$n\$ trials. In this article we conduct a systematic study of \$N_n\$ by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length \$k\$, the "non-overlapping", the "overlapping" and the "at least" scheme. In the special case of i.i.d. trials several new results are established.

Key words and phrases:    Binomial/negative binomial distribution of order \$k\$, success runs, Markov chain, probability generating function, probability mass function, moments.

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